On the asymptotic Plateau problem for CMC hypersurfaces in hyperbolic space

TL;DR

This paper proves the existence of constant mean curvature hypersurfaces in hyperbolic space with prescribed asymptotic boundary data, extending previous results to a broader class of boundary conditions.

Contribution

It establishes the existence of complete, properly embedded CMC hypersurfaces in hyperbolic space with bounded Euclidean graph boundary data, generalizing prior work on radial graphs.

Findings

Existence of CMC hypersurfaces with given boundary at infinity.

Construction of hypersurfaces for |H|<1.

Extension of previous radial graph results.

Abstract

Let $\mathbb{R}_{+}^{n+1}$ \ be the half-space model of the hyperbolic space $\mathbb{H}^{n+1}.$ It is proved that if $\Gamma\subset\left\{ x_{n+1}=0\right\} \subset\partial_{\infty}\mathbb{H}^{n+1}$ is a bounded $C^{0}$ Euclidean graph over $\left\{ x_{1}=0,\text{ }x_{n+1}=0\right\} $ then, given $\left\vert H\right\vert <1,$ there is a complete, properly embedded, CMC $H$ hypersurface $\Sigma$ of $\mathbb{H}^{n+1}$ such that $\partial_{\infty }S=\Gamma\cup\left\{ x_{n+1}=+\infty\right\} .$ This result can be seen as a limit case of the existence theorem proved by B. Guan and J. Spruck in \cite{GS} on CMC $\left\vert H\right\vert <1$ radial graphs with prescribed $C^{0}$ asymptotic boundary data.